28(16)
ſtruction:)
hence by ſubſtraction BK = KG + BF, and by ſubtraction again
FK = KG.
FK = KG.
Case 5th.
Suppoſe the given circle A to include B, and it be required that
the circles to be deſcribed be touched outwardly by A and inwardly by B.
the circles to be deſcribed be touched outwardly by A and inwardly by B.
Then let AB cut the circumferences in C and D, P and O:
and biſecting
CO in I, and ſetting off from I towards P, IL = the difference of the ſemidia-
meters of the given circles, and with A and B foci and IL tranſverſe axis de-
ſcribing an ellipſe LKI, it will be the locus of the centers of the circles deſcribed,
and the demonſtration, mutatis mutandis, is the ſame as in the laſt caſe.
CO in I, and ſetting off from I towards P, IL = the difference of the ſemidia-
meters of the given circles, and with A and B foci and IL tranſverſe axis de-
ſcribing an ellipſe LKI, it will be the locus of the centers of the circles deſcribed,
and the demonſtration, mutatis mutandis, is the ſame as in the laſt caſe.
Cases 6th and 7th.
Suppoſe the two given circles cut each other, and it be
required that the circles to be deſcribed either be touched and included in them
both, or be touched by them both and at the ſame time include them both.
required that the circles to be deſcribed either be touched and included in them
both, or be touched by them both and at the ſame time include them both.
These two caſes are ſimilar to caſes 1ſt and 2d, and as there, ſo alſo here,
the tranſverſe axis of the two oppoſite hyperbolas, which are the loci required,
muſt be taken = the difference of the ſemidiameters of the given circles. The
demonſtration is ſo alike, it need not be repeated.
the tranſverſe axis of the two oppoſite hyperbolas, which are the loci required,
muſt be taken = the difference of the ſemidiameters of the given circles. The
demonſtration is ſo alike, it need not be repeated.
PROBLEM IV.
Having a given point A, and a given right line BC, to determine the locus
of the centers of thoſe circles which ſhall paſs through A and touch BC.
of the centers of thoſe circles which ſhall paſs through A and touch BC.
From A draw AG perpendicular to BC, then with focus A and directrix BC
let a parabola be deſcribed, and it will be the locus required; for by the propert
of the curve FA always equals FG drawn perpendicular to the directrix.
let a parabola be deſcribed, and it will be the locus required; for by the propert
of the curve FA always equals FG drawn perpendicular to the directrix.
PROBLEM V.
Having a given point A, and a given circle whoſe center is B, to determine
the locus of the centers of all thoſe circles, which paſs through A, and at the
ſame time are touched by the given circle.
the locus of the centers of all thoſe circles, which paſs through A, and at the
ſame time are touched by the given circle.
Cases 1ſt and 2d.
Suppoſe the point A to lie out of the given circle, and
it be required that the circles to be deſcribed be either touched outwardly by the
given circle, or inwardly by it.
it be required that the circles to be deſcribed be either touched outwardly by the
given circle, or inwardly by it.
Let AB be drawn, and let it cut the given circumference where it is convex
towards A in the point C, and where it is concave in the point O: then biſecting
AC in E, and ſetting off from E towards B, EH = BC the given radius, and
with A and B foci and EH tranſverſe Axis deſcribing two oppoſite Hyperbolas
KEK and LHL, it is evident that KEK will be the locus of the centers of thoſe
circles which paſs thrugh A and are touched outwardly by the given circle, and
LHL will be the locus of the centers of thoſe circles which paſs through A and
are touched inwardly by the given circle.
towards A in the point C, and where it is concave in the point O: then biſecting
AC in E, and ſetting off from E towards B, EH = BC the given radius, and
with A and B foci and EH tranſverſe Axis deſcribing two oppoſite Hyperbolas
KEK and LHL, it is evident that KEK will be the locus of the centers of thoſe
circles which paſs thrugh A and are touched outwardly by the given circle, and
LHL will be the locus of the centers of thoſe circles which paſs through A and
are touched inwardly by the given circle.